If a substantial (though admittedly I have no idea how large) part of the natural rate of unemployment is caused by the job search, do more efficient means of communication create more efficient labor markets, and if so, can anyone quantify their impact with data?
Kroft and Pope (working paper, published in JoLE 2014) ask exactly this question, and their tentative answer is "no".
They view Craigslist as a unique opportunity to study the benefits of online job sites, since it grew rapidly and somewhat idiosyncratically while other popular sites grew steadily (leaving few opportunities for identification). That said, they don't have a great identification strategy - they just treat the rise of Craigslist starting in 2005 as an event study and compare MSAs where it grew quickly to MSAs where it grew sluggishly, making the case via various alternative specifications and falsification tests that there isn't any substantial bias.
Interestingly, they find that Craigslist does seem to have crowded out traditional newspaper classified ads, but had no discernible effect on unemployment. (This doesn't surprise me too much, since Craigslist seems like a pretty minor advance over traditional classifieds for job search, and if anything you'd expect a contribution from other sites that are more specialized or make more efforts to facilitate matching.)
On the other hand, they find that Craigslist does seem to have some effect on rental vacancy rates. Perhaps simple, searchable Craigslist-style classified ads facilitate rental search (I know they have for me) but are too crude to do much for employment.